In a triangle with angles measuring $a$, $b$ and $c$ degrees, the mean of $b$ and $c$ is $a$. What is the value of $a$?
Solution: We know that $a+b+c=180^\circ$ and we are given that $\frac{b+c}{2}=a$. Thus, we have $b+c=2a$. We can plug this into $a+b+c=180^{\circ}$ to get $3a = 180^{\circ}$, which means $a=\boxed{60^\circ}$.